In this subsection we consider the intermediate temperature limit i.e. \(l^{\prime }T \ll 1\ll lT \)  at the high energy limit i.e i.e. \(l\Lambda _c\ \&\ l^{\prime }\Lambda _c\ll 1\) . In this limit \(\Lambda _c \ll T\)  (\(r_c \ll r_H\) ), see figure REF . In this regime, the extremal surface \(\Gamma _{l^{\prime }}\)  is restricted to be near the AdS boundary and the turning point of the extremal surface \(\Gamma _{l^{\prime }+2l}\)  approaches \(r_H\) . Doing some calculations and keeping up to the 4th order in \(l^{\prime }\Lambda _c\)  and \(l^{\prime }T\) , we obtain the following expression (the details of the calculation can be found in appendix )

Ep(l'c,l'T)4GN(5)Ep(l',l'c,l'T)L2c T=1(l'c)(l'T)2a12-C34(l'T)4        +[C7-34(l'T)+9128(l'cl'T)2-C4(l'T)4
](l' c)2        -[C14+C5(l'T)4](l'c)4,   C1,C4>0, C3,C5,C7<0,    
where \(\tilde{E}_p(l^{\prime }\Lambda _c,l^{\prime }T)\)  is the redefined \(E_p(l^{\prime },l^{\prime }\Lambda _c,l^{\prime }T)\)  which is given by eq.(REF ) and the numerical constant \(C_7\)  is given by eq.(REF ) in appendix . The first and second terms are the same as in the previous subsection. It is seen that \(\tilde{E}_p(l^{\prime }\Lambda _c,l^{\prime }T)\)  does not depend on the length of the subsystems and diverges in \(l^{\prime }\rightarrow 0\)  limit.
Since \(C_3\)  is a negative constant the second term, the dominant term, is always positive and hence the finite temperature correction increases \(\tilde{E}_p(l^{\prime }\Lambda _c,l^{\prime }T)\) . The logarithmic term is the dominant term in the non-conformal effects. This terms is always positive and hence the non-conformal effects increase \(\tilde{E}_p(l^{\prime }\Lambda _c,l^{\prime }T)\) . For \(\Lambda _c=0\)  we reproduce the previous results obtained for AdS block holes [1]}.
From eq.(REF ) we observe that \(\tilde{E}_p(l^{\prime }\Lambda _c,l^{\prime }T)\)  decreases by increasing \(l^{\prime }\Lambda _c \)  (\(l^{\prime }T\) ) for fixed \(l^{\prime }T\)  (\(l^{\prime }\Lambda _c \) ).
